Pell equation and randomness

Research output: Contribution to journalArticle

Abstract

Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, located along a straight line passing through the origin with a quadratic irrational slope. The case of inhomogeneous Pell inequalities it is equivalent to finding the integer lattice points in translated copies of a long and narrow tilted hyperbolic region with a quadratic irrational slope. We prove randomness in these natural lattice point counting problems. We have two main results: a central limit theorem (Theorem 2.1) and a law of the iterated logarithm (Theorem 2.2). The classical Circle Problem (counting lattice points in large circles) is a notoriously difficult open problem. What our results show is that the hyperbolic analog of the Circle Problem is solvable with striking precision.

Original languageEnglish (US)
Pages (from-to)1-108
Number of pages108
JournalPeriodica Mathematica Hungarica
Volume70
Issue number1
DOIs
StatePublished - Jan 1 2015
Externally publishedYes

Fingerprint

Pell's equation
Lattice Points
Randomness
Counting Problems
Integer Points
Circle
Slope
Law of the Iterated Logarithm
Theorem
Straight Line
Central limit theorem
Open Problems
Analogue
Integer

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Central limit theorem
  • Discrepancy
  • Lattice point counting in specified regions
  • Law of the iterated logarithm

Cite this

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Pell equation and randomness. / Beck, József.

In: Periodica Mathematica Hungarica, Vol. 70, No. 1, 01.01.2015, p. 1-108.

Research output: Contribution to journalArticle

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